Bionic robots, particularly those inspired by primate locomotion, have garnered significant interest in robotics research due to their ability to perform complex, dynamic movements such as brachiation. Brachiation, the arm-swinging motion used by primates like gibbons, involves two primary modes: continuous brachiation (swinging) and ricochetal brachiation (flight). This paper focuses on ricochetal brachiation, where the bionic robot releases one support, flies through the air, and grasps another target. The challenge lies in the tight coupling of locomotion modes and the underactuated dynamics of the system. Here, we present a comprehensive trajectory planning and control strategy for a two-link bionic robot to achieve ricochetal brachiation, emphasizing the use of virtual constraints and segmented motion planning.
The bionic robot model consists of two links representing the arms of a primate, with an underactuated joint between them. This design mimics the dynamic behavior of primates during brachiation. The motion is divided into three phases: swing phase (pre-flight), free-flight phase, and landing phase (post-flight). Each phase involves distinct dynamical models and control requirements. The key is to plan trajectories that ensure smooth transitions between phases while meeting target constraints, such as flight distance and energy efficiency. This bionic robot approach enables adaptive and flexible motion planning for various environments.
In the swing phase, the bionic robot is attached to a support point with one end, forming an underactuated system with two degrees of freedom. The dynamics are derived using Lagrange’s equations. Let $m_1$ and $m_2$ be the masses of the upper arm (link 1) and lower arm (link 2), respectively. The lengths are $l_1$ and $l_2$, with distances to centers of mass $l_{c1}$ and $l_{c2}$. The angles $\theta_1$ and $\theta_2$ define the configuration, where $\theta_1$ is the angle between link 1 and the vertical direction, and $\theta_2$ is the angle between link 2 and the extension of link 1. The input torque $\tau$ acts between the links. The equations of motion are:
$$ d_{11} \ddot{\theta}_1 + d_{12} \ddot{\theta}_2 + h_1 + \phi_1 = 0, $$
$$ d_{21} \ddot{\theta}_1 + d_{22} \ddot{\theta}_2 + h_2 + \phi_2 = \tau, $$
where the coefficients are defined as:
$$ d_{11} = \omega_1 + \omega_2 + 2\omega_3 \cos \theta_2, \quad d_{22} = \omega_2, $$
$$ d_{12} = d_{21} = \omega_2 + \omega_3 \cos \theta_2, $$
$$ \phi_1 = \omega_4 g \sin \theta_1 + \omega_5 g \sin(\theta_1 + \theta_2), $$
$$ \phi_2 = \omega_5 g \sin(\theta_1 + \theta_2), $$
$$ h_1 = -\omega_3 \dot{\theta}_2 (2 \dot{\theta}_1 + \dot{\theta}_2) \sin \theta_2, \quad h_2 = \omega_3 \dot{\theta}_1^2 \sin \theta_2, $$
with parameters:
$$ \omega_1 = I_1 + m_1 l_{c1}^2 + m_2 l_1^2, \quad \omega_2 = I_2 + m_2 l_{c2}^2, $$
$$ \omega_3 = m_2 l_1 l_{c2}, \quad \omega_4 = m_1 l_{c1} + m_2 l_1, \quad \omega_5 = m_2 l_{c2}. $$
Here, $I_1$ and $I_2$ are moments of inertia, and $g$ is gravity. This model captures the underactuated nature of the bionic robot during swinging, where only one joint is actuated.
In the free-flight phase, the bionic robot releases all supports and moves under gravity alone. The system becomes a floating base with conservation of angular momentum. Let $\theta_{c0}$ be the angle between the line from the grasp point to the center of mass (COM) and the vertical at take-off, $l_{c0}$ the distance from COM to grasp point, $\beta$ the angle of COM velocity relative to horizontal, $d$ the flight distance, and $R$ the horizontal displacement of COM. The angular momentum $H$ is conserved:
$$ H = c_1 [(c_2 + c_3 \cos \theta_2) \dot{\theta}_1 + (c_4 + c_5 \cos \theta_2) \dot{\theta}_2] = H_0, $$
where constants are:
$$ c_1 = \frac{1}{m_1 + m_2}, \quad c_2 = m_1(I_1 + I_2) + m_2[I_1 + I_2 + m_1(l_1 – l_{c1})^2 + l_{c2}^2], $$
$$ c_3 = 2 m_1 m_2 (l_1 – l_{c1}) l_{c2}, \quad c_4 = m_1 m_2 l_{c2}^2 + (m_1 + m_2) I_2, $$
$$ c_5 = m_1 m_2 (l_1 – l_{c1}) l_{c2}. $$
This allows expressing $\dot{\theta}_1$ in terms of $\theta_2$ and $\dot{\theta}_2$:
$$ \dot{\theta}_1 = \frac{H_0}{c_1 (c_2 + c_3 \cos \theta_2)} – \frac{c_4 + c_5 \cos \theta_2}{c_2 + c_3 \cos \theta_2} \dot{\theta}_2. $$
Thus, the bionic robot can adjust its posture during flight by controlling $\theta_2$ and $\dot{\theta}_2$ via torque $\tau$.
Trajectory planning for ricochetal brachiation involves determining pre-flight and post-flight conditions based on target constraints. For a given flight distance $d$, we compute take-off and landing姿态 parameters. Assume symmetry in the flight path. Let take-off instant variables be $\theta_1 = \theta_{10}$, $\theta_2 = \theta_{20}$, $\dot{\theta}_1 = \dot{\theta}_{10}$, $\dot{\theta}_2 = \dot{\theta}_{20}$, and landing instant variables be $\theta_1 = \theta_{1f}$, $\theta_2 = \theta_{2f}$, $\dot{\theta}_1 = \dot{\theta}_{1f}$, $\dot{\theta}_2 = \dot{\theta}_{2f}$. The COM coordinates at take-off are:
$$ x_{c0} = l_{c0} \sin \theta_{c0} = \frac{m_1}{M} l_{c1} \sin \theta_{10} + \frac{m_2}{M} [l_1 \sin \theta_{10} + l_{c2} \sin(\theta_{10} + \theta_{20})], $$
$$ y_{c0} = -l_{c0} \cos \theta_{c0} = -\frac{m_1}{M} l_{c1} \cos \theta_{10} + \frac{m_2}{M} [-l_1 \cos \theta_{10} – l_{c2} \cos(\theta_{10} + \theta_{20})], $$
where $M = m_1 + m_2$. The COM velocity components are:
$$ v \cos \beta = \dot{x}_{c0} = \frac{m_1}{M} l_{c1} \dot{\theta}_{10} \cos \theta_{10} + \frac{m_2}{M} [l_1 \dot{\theta}_{10} \cos \theta_{10} + l_{c2} (\dot{\theta}_{10} + \dot{\theta}_{20}) \cos(\theta_{10} + \theta_{20})], $$
$$ v \sin \beta = \dot{y}_{c0} = \frac{m_1}{M} l_{c1} \dot{\theta}_{10} \sin \theta_{10} + \frac{m_2}{M} [l_1 \dot{\theta}_{10} \sin \theta_{10} + l_{c2} (\dot{\theta}_{10} + \dot{\theta}_{20}) \sin(\theta_{10} + \theta_{20})]. $$
From symmetry, the horizontal displacement $R$ satisfies $2 l_{c0} \sin \theta_{c0} + R = d$. The range equation gives $R = (v^2 \sin 2\beta)/g$, and flight time $t_f = (2v \sin \beta)/g$. For landing, similar equations hold with respect to the new grasp point. By imposing an additional constraint, such as $\beta = \text{constant}$, we can solve for all unknown variables. This flexibility allows energy adjustment for the bionic robot. Table 1 summarizes planning values for different flight distances $d$ and take-off angles $\beta$, showcasing how energy $E$ varies.
| $d$ (m) | $\beta$ (rad) | $v$ (m/s) | $E$ (J) | $t_f$ (s) |
|---|---|---|---|---|
| 2.4 | 0.5 | 4.3348 | 11.7317 | 0.4237 |
| 2.4 | 0.6 | 3.8565 | 5.9708 | 0.4439 |
| 2.4 | 0.7 | 3.4824 | 2.3887 | 0.4574 |
| 2.5 | 0.5 | 4.4140 | 11.6483 | 0.4314 |
| 2.5 | 0.6 | 3.9348 | 6.0369 | 0.4529 |
| 2.5 | 0.7 | 3.6563 | 2.6285 | 0.4679 |
| 2.6 | 0.5 | 4.4959 | 11.7309 | 0.4394 |
| 2.6 | 0.6 | 4.0165 | 6.2615 | 0.4624 |
| 2.6 | 0.7 | 3.6478 | 3.0257 | 0.4791 |
| 2.7 | 0.5 | 4.5803 | 11.9647 | 0.4477 |
| 2.7 | 0.6 | 4.1011 | 6.6284 | 0.4721 |
| 2.7 | 0.7 | 3.7364 | 3.5613 | 0.4907 |
To achieve the pre-flight姿态, we employ virtual-constraint-based trajectory planning. A virtual constraint relates the angles, e.g., $\theta_1 = \varphi(\theta_2) = a \theta_2 + b$, where $a$ and $b$ are determined from target conditions. Substituting into the dynamics yields the zero dynamics:
$$ \alpha(\theta_2) \ddot{\theta}_2 + \sigma(\theta_2) \dot{\theta}_2^2 + \gamma(\theta_2) = 0, $$
with coefficients:
$$ \alpha(\theta_2) = \frac{\partial \varphi}{\partial \theta_2} d_{11} + d_{12}, $$
$$ \sigma(\theta_2) = \frac{\partial^2 \varphi}{\partial \theta_2^2} d_{11} – \omega_3 \sin \theta_2 \left(2 \frac{\partial \varphi}{\partial \theta_2} + 1\right), $$
$$ \gamma(\theta_2) = \omega_4 g \sin(\varphi(\theta_2)) + \omega_5 g \sin(\varphi(\theta_2) + \theta_2). $$
Let $Y = \dot{\theta}_2^2$. The zero dynamics can be integrated to obtain the orbit equation:
$$ Y = \dot{\theta}_2^2 = I^{-1} \int_{\theta_{2d}}^{\theta_2} I \gamma_0(x) dx + Y_0 I^{-1}, $$
where $I = \exp\left( \int_{\theta_{2d}}^{\theta_2} \sigma_0(x) dx \right)$, $\sigma_0(\theta_2) = 2\sigma(\theta_2)/\alpha(\theta_2)$, and $\gamma_0(\theta_2) = -2\gamma(\theta_2)/\alpha(\theta_2)$. This defines the desired trajectory for the bionic robot.
Control strategy involves stabilizing two outputs: the virtual constraint $h = \varphi(\theta_2) – \theta_1$ and the orbit function $U = \dot{\theta}_2^2 – Y(\theta_2)$. A Lyapunov-based controller is designed. The control torque is:
$$ \tau = \frac{-\chi – f_1 – K_1 \dot{h} – K_2 h + \frac{\partial \varphi}{\partial \theta_2} f_2 + \frac{\partial^2 \varphi}{\partial \theta_2^2} \dot{\theta}_2^2}{g_1 – \frac{\partial \varphi}{\partial \theta_2} g_2}, $$
where $g_1 = -d_{12} / (d_{11} d_{22} – d_{12} d_{21})$, $g_2 = d_{11} / (d_{11} d_{22} – d_{12} d_{21})$, $f_1$ and $f_2$ are terms from dynamics, and $K_1$, $K_2$ are gains. The virtual input $\chi$ is:
$$ \chi = \frac{K_3 \arctan\left( \frac{U \dot{\theta}_2}{\alpha} \right) – \sigma U + f_h(\theta_2, h) + f_{\dot{h}}(\theta_2, \dot{\theta}_2, \dot{h})}{d_{11}}, $$
with $f_h$ and $f_{\dot{h}}$ as coupling terms. Gains are chosen such that $K_1 > \max\left| \frac{2\omega_3 \dot{\theta}_2 \sin \theta_2}{d_{11}} \right|$ and $K_2 > \max\left| \frac{-\omega_4 g \sin(\varphi(\theta_2)) + \omega_5 g \sin(\varphi(\theta_2) + \theta_2)}{d_{11}} \right|$. This ensures convergence to the target trajectory for the bionic robot.
During free-flight, the bionic robot controls $\theta_2$ with a constant rate $\dot{\theta}_{2k}$ to maintain desired posture. The switching control integrates swing, flight, and landing phases. A schematic of the switch controller ensures seamless transitions based on boundary conditions.
Simulation experiments validate the approach. Parameters for the bionic robot are: $m_1 = m_2 = 1 \, \text{kg}$, $l_1 = l_2 = 1 \, \text{m}$, $l_{c1} = l_{c2} = 0.5 \, \text{m}$, $I_1 = I_2 = 1/12 \, \text{kg} \cdot \text{m}^2$, $g = 9.81 \, \text{m/s}^2$, and $l_{c0} = 0.982 \, \text{m}$. For $d = 2.5 \, \text{m}$ and $\beta = 0.7 \, \text{rad}$, pre-flight conditions are $\theta_{10} = 0.5648$, $\theta_{20} = 0.4397$, $\dot{\theta}_{10} = 3.3461$, $\dot{\theta}_{20} = 1.1666$. The virtual constraint is $\theta_1 = 2.8682 \theta_2 – 0.6965$. With gains $K_1 = 20$, $K_2 = 400$, $K_3 = 1000$, and initial state $[\theta_{1d}, \theta_{2d}, \dot{\theta}_{1d}, \dot{\theta}_{2d}] = [-\pi/2, -0.3, 0, 0]$, the bionic robot achieves the target posture at $t = 0.9 \, \text{s}$ and periodically thereafter, enabling take-off at any cycle. The flight phase shows COM trajectory matching parabolic motion. Similarly, for $d = 2.7 \, \text{m}$ and $\beta = 0.5 \, \text{rad}$, pre-flight conditions are $\theta_{10} = 0.3677$, $\theta_{20} = 0.4397$, $\dot{\theta}_{10} = 4.3414$, $\dot{\theta}_{20} = 1.3355$, with virtual constraint $\theta_1 = 3.2507 \theta_2 – 1.0617$. The controller successfully drives the bionic robot to the desired states.
Table 2 summarizes key parameters and outcomes for these simulations, highlighting the adaptability of the bionic robot.
| Parameter | Case 1: $d=2.5 \, \text{m}$, $\beta=0.7$ | Case 2: $d=2.7 \, \text{m}$, $\beta=0.5$ |
|---|---|---|
| Pre-flight $\theta_{10}$ (rad) | 0.5648 | 0.3677 |
| Pre-flight $\theta_{20}$ (rad) | 0.4397 | 0.4397 |
| Pre-flight $\dot{\theta}_{10}$ (rad/s) | 3.3461 | 4.3414 |
| Pre-flight $\dot{\theta}_{20}$ (rad/s) | 1.1666 | 1.3355 |
| Virtual constraint $a$ | 2.8682 | 3.2507 |
| Virtual constraint $b$ | -0.6965 | -1.0617 |
| Control gain $K_1$ | 20 | 20 |
| Control gain $K_2$ | 400 | 400 |
| Control gain $K_3$ | 1000 | 1000 |
| Time to target (s) | 0.9 | ~1.0 |
| Flight time $t_f$ (s) | 0.4679 | 0.4477 |
The bionic robot’s performance demonstrates effective trajectory planning and control. The use of virtual constraints allows precise regulation of underactuated dynamics, while the segmented planning accommodates varying flight distances. Energy efficiency can be tuned by selecting $\beta$, as shown in Table 1. This adaptability is crucial for real-world applications where environmental constraints vary.
Further analysis involves examining the robustness of the bionic robot to disturbances. Perturbations in initial conditions or model parameters can be mitigated by adaptive control schemes. For instance, online adjustment of virtual constraints based on sensor feedback could enhance the bionic robot’s reliability. Additionally, incorporating more complex models, such as adding flexibility or additional degrees of freedom, would extend the bionic robot’s capabilities. The principles here can be applied to other bionic robots inspired by animal locomotion.
In conclusion, this paper presents a comprehensive framework for ricochetal brachiation of a two-link bionic robot. The trajectory planning method flexibly determines pre-flight and post-flight conditions, enabling adaptive motion for different distances. The control strategy, based on virtual constraints and Lyapunov stability, ensures accurate tracking and phase transitions. Simulations validate the approach, showing successful flight and landing. Future work includes experimental implementation on a physical bionic robot platform and extension to continuous brachiation with multiple cycles. The bionic robot paradigm offers promising avenues for advanced robotic mobility in complex environments.
The bionic robot’s design emphasizes biomimicry, drawing from primate brachiation to achieve dynamic, energy-efficient motion. By leveraging underactuated dynamics and virtual constraints, the bionic robot can perform sophisticated maneuvers that mimic natural behaviors. This research contributes to the broader field of bionic robotics, where inspiration from biology leads to innovative engineering solutions. As bionic robots evolve, they may find applications in search and rescue, inspection, and entertainment, showcasing the power of nature-inspired design.